Emergent rotational order and reentrant global order of Vicsek agents in a complex noise environment

Noisy pursuit in complex environments drives emergent collective behaviors in active matter systems. A compelling platform to study the impact of environment cues is provided by the standard Vicsek model for studying flocking and swarming phenomena. In this study, we explore the collective dynamics of Vicsek agents in a complex noise environment, featuring a noiseless circular region ($η_{\text{c}} = 0.0$) surrounded by a noisy outer region ($η_{\text{b}} = 1.0$, tunable), with a mutually repelling interactions. By varying the outer noise intensity, we observe an emergent rotational order ($ϕ_r$) that peaks at higher noise levels ($η_{\text{b}}\sim 1$), as revealed by phase and susceptibility plots. Global order follows ($ϕ$) follows a `U' shaped curve, $ϕ\sim 0.965$ at $η_b=0$, dies down to $ϕ\sim 0.57$ at $η_b=0.9$ and re-enters at $η_b > 1$ and peaks $ϕ\sim 0.960$ at $η_b=1.5$. The latter rise attributing to $ϕ_r$ increase. Higher particle velocities enhance escape rates ($κ$) from the circular region, with slower-moving agents exhibiting greater virtual confinement. We quantify escape dynamics through time-averaged and first-passage escape rates, demonstrating velocity-dependent retention on the probability of finding the bi-motility agent flocks at a give time resulting in segregation and trapping. Introducing a gradual noise increase from the circle's center to the outer region reduces both global ($ϕ$) and rotational ($ϕ_r$) order, underscoring the impact of environmental heterogeneity and sudden annealing over gradual change. These findings offer insights into predicting and manipulating active agent dynamics in heterogeneous environments, with applications in biological and synthetic swarming systems.